Categorification of the colored $\mathfrak{sl}_3$-invariant

TL;DR

This paper explicitly constructs resolutions of simple modules over quantum sl_3 and uses them to categorify the colored sl_3-invariant of framed links through complex structures of graded modules.

Contribution

It provides explicit resolutions for all finite-dimensional simple U_q(sl_3)-modules and applies these to categorify the colored sl_3-invariant of framed links.

Findings

Resolutions of all simple U_q(sl_3)-modules are explicitly constructed.

Categorification of the colored sl_3-invariant is achieved using complexes of graded modules.

The approach links module resolutions to link invariants in a novel way.

Abstract

We give explicit resolutions of all finite dimensional, simple $U_q(\mathfrak{sl_3})$-modules. We use these resolutions to categorify the colored $\mathfrak{sl}_3$-invariant of framed links via a complex of complexes of graded $\mathbb{Z}$-modules.